Algebra, Geometry, and Combinatorics Seminar

Title: The solution space of an A-hypergeometric system

Abstract: An A-hypergeometric system is a parametric system of PDEs arising from a toric ideal. I will survey the story of how its solution space varies with the parameters, which takes surprising turns into the behavior of semigroups, local cohomology, and homological algebra.

Prerequisites: It may be helpful to have a basic understanding of semigroup rings.

Title: Hirota Varieties

Abstract: The Kadomtsev-Petviashvili (KP) equation is a differential equation whose study yields interesting connections between integrable systems and algebraic geometry. In this talk I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes all KP solutions arising from such a sum. I will then discuss an ongoing work, studying the Hirota variety of the g-cube, which is the Delaunay polytope associated to an irreducible rational nodal curve with g nodes.

Prerequisites: some familiarity with basic algebraic geometry (in particular, what is a variety?). Deeper knowledge of algebraic and tropical geometry would help to fully understand the deeper parts of the talk, but I will try to make the big ideas of the talk accessible to everyone.

Title: Dyson's Crank and the Mex Partition Statistics

Abstract: A young Freeman Dyson discovered the rank of a partition, a simple statistic which provided combinatorial perspectives on some congruences found by Ramanujan. But one congruence needed something else that he could not find, but named anyway: the crank. It was finally found more than 40 years later by George Andrews and Frank Garvan. The definition of the crank is a bit tricky, but it has gone on to become a major topic in the study of partitions. A recent statistic is the minimal excluded part of a partition, known as the mex, borrowing a term from combinatorial game theory. This simpler parameter has surprising connections to the crank and is more amenable to combinatorial considerations. This is joint work with James Sellers, Dennis Stanton, and Ae Ja Yee.

Prerequisites: There will be q-series (analytic) arguments, but I will emphasize combinatorial approaches most of the time.

Title: Logarithmic Voronoi cells

Abstract: Given a point in a statistical model, the fiber of the MLE map corresponding to that point is known as its logarithmic Voronoi cell. For discrete models, each logarithmic Voronoi cell lives inside its log-normal polytope, and the logarithmic Voronoi cells corresponding to all points in the model fill the probability simplex. I will introduce these notions and investigate when logarithmic Voronoi cells are polytopes and when they are non-polytopal convex sets. I will draw special attention to the linear models, investigate the combinatorial type of logarithmic Voronoi cells in this case, and how it degenerates on the boundary of the simplex. I will also talk about the Gaussian case, where logarithmic Voronoi cells fill the cone of positive definite matrices. I will discuss the structure of these cells for different families of Gaussian models, e.g. concentration and correlation models. This talk is based on joint work with Alex Heaton and work-in-progress with Serkan Hoşten.

Prerequisites: some background in polyhedral and convex geometry would be helpful.

Title: ∞-category theory for undergraduates

Abstract: At its current state of the art, ∞-category theory is challenging to explain even to specialists in closely related mathematical areas. Nevertheless, historical experience suggests that in, say, a century's time, we will routinely teach this material to undergraduates. This talk describes one dream about how this might come about --- under the assumption that 22nd century undergraduates have absorbed the background intuitions of homotopy type theory/univalent foundations.

Prerequisites: I'll assume that type theory and homotopy type theory are totally unfamiliar. It would be helpful, though, to have seen a couple of concepts from point set topology however: eg path, homotopy, homotopy equivalence, contractible space. These aren't so much prerequisites but things you might google if you were so inclined to google a few terms before attending a talk.

Title: Rational surfaces and locally recoverable codes

Abstract: Motivated by large-scale storage problems around data loss, a budding branch of coding theory has surfaced in the last decade or so, centered around locally recoverable codes. These codes have the property that individual symbols in a codeword are functions of other symbols in the same word. If a symbol is lost (as opposed to corrupted), it can be recomputed, and hence a code word can be repaired. Algebraic geometry has a role to play in the design of codes with locality properties. In this talk I will explain how to use algebraic surfaces to both reinterpret constructions of optimal codes already found in the literature, and to find new locally recoverable codes, many of which are optimal (in a suitable sense). This is joint work with Cecília Salgado and Felipe Voloch.

Prerequisites: Introduction to coding theory and bound on codes. A good reference for those is the first chapter of "Codes and Curves" from Judy L. Walker: link

Title: Metric Algebraic Geometry

Abstract: A real algebraic variety is the set of points in real Euclidean space that satisfy a system of polynomial equations. Metric algebraic geometry is the study of properties of real algebraic varieties that depend on a distance metric. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, and the reach of an algebraic variety. We also show applications to the computational study of the geometry of data with nonlinear models.

Prerequisites: Background in linear algebra and point-set topology is helpful. It will also be helpful if you have been introduced to the concept of a real algebraic variety (as the zero set of a system of polynomial equations) as well as properties such as its dimension and degree, but I will quickly review these concepts.

Title: Log-concave graphical models

Abstract: The problem of density estimation aims to find an unknown density p on R^d given i.i.d. samples from it. In this work we study how to solve this problem assuming that p is log-concave and obeys certain dependence structure — that of an undirected graphical model. This is an instance of “nonparametric density estimation”, a challenging problem in theoretical statistics. We show that the maximum likelihood estimator (MLE) for p exists and is unique with probability 1. Furthermore, we precisely describe the support of the MLE, and we find a finite dimensional convex optimization problem that computes the MLE.

This is based on joint work with Kaie Kubjas, Olga Kuznetsova, Pardis Semnani, and Luca Sodomaco

Prerequisites: Maximum likelihood estimation

Title: A complete combinatorial interpretation of Goncarov polynomials.

Abstract: Classical Goncarov polynomials arose in numerical analysis as a basis for the solutions of the Goncarov interpolation problem.

These polynomials provide a natural algebraic tool in the enumerative theory of parking functions. Parking functions are combinatorial objects which were introduced in 1966 by Konheim and Weiss. They have been well-studied in the literature due to their numerous connections and have several generalizations and extensions. In this talk, we present a complete combinatorial interpretation for any sequence of generalized Goncarov polynomials realized in the lattice of partitions. This is joint work with Catherine Yan.

Prerequisites: Discrete Mathematics course or Combinatorics

Title: The Topological Symmetry Groups of the Heawood Graph

Abstract: Although, motivated by chemistry, spatial graph theory has now become a subfield of low-dimensional topology closely related to knot theory. In particular, the study of topological symmetry groups of graphs embedded in the 3-sphere can be thought of as a generalization of the study of symmetries of knots and links. For a given embedding, we are interested in the automorphisms of the graph that are induced by a homeomorphism of the 3-sphere. This subgroup of the automorphism group of the graph is known as the topological symmetry group of that embedding. We will discuss recent results classifying which groups can occur as the topological symmetry group of some embedding of the Heawood graph in the 3-sphere.

Prerequisites: should be fairly accessible to students with a background in algebra, and maybe a little topology